On-shell methods for off-shell quantities in N = 4 Super Yang-Mills: From scattering amplitudes to form factors and the dilatation operator

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Abstract

Planar maximally supersymmetric Yang-Mills theory (N = 4 SYM) is a special quantum
fi eld theory. A few of its remarkable features are conformal symmetry at the quantum
level, evidence of integrability and, moreover, it is a prime example of the AdS/CFT duality.
Triggered by Witten's twistor string theory [1], the past 15 years have witnessed
enormous progress in reformulating this theory to make as many of these special features
manifest, from the choice of convenient variables to recursion relations that allowed new
mathematical structures to appear, like the Grassmannian [2]. These methods are collectively
referred to as on-shell methods. The ultimate hope is that, by understanding
N = 4 SYM in depth, one can learn about other, more realistic quantum fi eld theories.
The overarching theme of this thesis is the investigation of how on-shell methods can aid
the computation of quantities other than scattering amplitudes. In this spirit we study
form factors and correlation functions, said to be partially and completely off-shell quantities,
respectively. More explicitly, we compute form factors of half-BPS operators up
to two loops, and study the dilatation operator in the SO(6) and SU(2j3) sectors using
techniques originally designed for amplitudes. A second part of the work is dedicated
to the study of scattering amplitudes beyond the planar limit, an area of research which
is still in its infancy, and not much is known about which special features of the planar
theory survive in the non-planar regime. In this context, we generalise some aspects of the
on-shell diagram formulation of Arkani-Hamed et al. [3] to take into account non-planar
corrections.